Determine whether the given function satisfies the...

Determine whether the given function satisfies the hypotheses of Rolle's Theorem on the indicated interval. If so, find all values of $c$ that satisfy the conclusion of the theorem.
$$
f(x)=\sin x ; \quad[-\pi, 2 \pi]
$$

Key Concepts

Finding Critical Points via the Derivative

Once the hypotheses of Rolle's Theorem are verified, the theorem guarantees the existence of at least one point where the derivative equals zero. Solving the equation f'(x) = 0 allows one to find these critical points. This process of setting the derivative equal to zero and solving for x is central to finding extremum points and analyzing function behavior in calculus.

Equal Endpoint Values

An essential hypothesis of Rolle's Theorem is that the function must have equal values at the endpoints of the interval, i.e., f(a) = f(b). This condition ensures that the function, which starts and ends at the same height, must have at least one point in between where the instantaneous rate of change (the derivative) is zero, indicating a horizontal tangent.

Continuity and Differentiability

For Rolle's Theorem to hold, the function must be continuous on the closed interval and differentiable on the open interval. Continuity ensures that there are no breaks or jumps in the function, while differentiability ensures that the function has a well-defined tangent (slope) at every point in the open interval. These conditions are key to applying many results in calculus, particularly those involving rates of change.

Rolle's Theorem

Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and takes equal values at the endpoints, then there exists at least one point c in (a, b) where the derivative of the function is zero. This concept is a fundamental result in calculus that guarantees the existence of stationary points under certain conditions.

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